## Suppression of aliasing in multi-sensor scanning absolute profile measurement

Optics Express, Vol. 17, Issue 13, pp. 11098-11106 (2009)

http://dx.doi.org/10.1364/OE.17.011098

Acrobat PDF (985 KB)

### Abstract

The task of anti-aliasing in absolute profile measurement by multi-sensor scanning techniques is considered. Simulation results are presented which demonstrate that aliasing can be highly reduced by a suitable choice of the scanning steps. The simulation results were confirmed by results obtained for interferometric measurements (Nyquist frequency 1/646 µm^{−1}) on a specifically designed chirp specimen with sinusoidal waves of amplitude 100 nm and wavelengths from 2.5 mm down to 19 µm.

© 2009 Optical Society of America

## 1. Introduction

1. L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. **33**, 7334–7338 (1994).
[CrossRef] [PubMed]

3. J. Cohen-Sabban and D. Reolon, “Vibration insensitive 3D-profilometry: a new type of white light interferometric microscopy” Proc. SPIE 7064, (2008). [CrossRef]

4. D J Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E Sci. Instrum. **9**, 531–536 (1976).
[CrossRef]

5. W. Gao and S. Kiyono, “High accuracy profile measurement of a machined surface by the combined method,” Measurement **19**, 55–64 (1996).
[CrossRef]

6. C. Elster, I. Weingaertner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. **30**, 32–38 (2006).
[CrossRef]

7. A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express **16**, 11975–11986 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975.
[CrossRef] [PubMed]

8. µPhase Interferometer, FISBA Optik AG, CH-9016 St. Gallen and FISBA Optik GmbH, Berlin, http://www.fisba.ch/.

9. MPLS 180, STIL - 595, rue Pierre Berthier - Domaine de Saint Hilaire - 13855 Aix en Provence Cedex 3 - FRANCE, http://www.stilsa.com/.

10. E. Marsh, J. Couey, and R. Vallance, “Nanometer-Level Comparison of Three Spindle Error Motion Separation Techniques,” J. Manuf. Sci. Eng. **128**, 180–187 (2006)
[CrossRef]

12. B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE **5878**, 587806 (2005).
[CrossRef]

## 2. Absolute profile measurement by a multi-sensor scanning technique

7. A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express **16**, 11975–11986 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975.
[CrossRef] [PubMed]

_{i}) and offset (a

_{i}) errors. Furthermore, each of the distance sensors shows a systematic offset error ε

_{j}. The topography can be reconstructed at the positions x

_{k}(k=0,1,…,K-1). It has been shown in [7

7. A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express **16**, 11975–11986 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975.
[CrossRef] [PubMed]

_{i}(offsets), b

_{i}(tilts), ε

_{j}(systematic sensor errors) and f(x

_{0}),…,f(x

_{K-1}) (topography). The additional tilt measurements are essential to ensure absolute profile reconstruction since otherwise the quadratic part of the topography is lost due to the unknown systematic sensor errors, cf. [6

6. C. Elster, I. Weingaertner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. **30**, 32–38 (2006).
[CrossRef]

_{s}=x

_{k}-x

_{k-1}has to be larger than half of the pixel distance d

_{pix}. The coefficients c

_{k}in equation (1) are based on a polynomial interpolation with a high degree o of the interpolation polynomial. The transfer function associated with this interpolation scheme is shown in Fig. 2.

**16**, 11975–11986 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975.
[CrossRef] [PubMed]

## 3. Aliasing – anti-aliasing

_{k}which form an equidistant grid with spacing d

_{s}. In order to subsequently allow for an exact interpolation between these grid points, the topography must not contain spatial frequencies beyond the Nyquist frequency f

_{Nyq}=1/2d

_{s}. The reason is that these higher frequencies would cause aliasing errors, i.e. they would emerge in such an interpolation as lower frequencies which are not present in the actual topography.

_{k}where the topography is reconstructed, by using d

_{step}=d

_{pix}. However, then the measurement positions are gathered at the reconstruction positions x

_{k}and no information on the topography between the positions x

_{k}is available. Consequently, for such proceeding aliasing cannot be controlled or reduced.

_{Nyq}are correctly reconstructed while higher frequencies are suppressed. Such behavior is ideal with respect to avoiding aliasing errors. However, Fig. 2 was calculated for continuously spaced measurement positions corresponding to an infinitesimal small scanning step d

_{step}. As an approximation to this, the measurement positions of the line sensor array should be distributed as uniformly as possible. This dense sampling of the topography is then expected to approximate the ideal behavior indicated in Fig. 2 and hence to suppress the aliasing effects which are obtained for sparse sampling. Dense sampling can be realized easily as follows. For a sensor head with N equidistant distance sensors, the scanning step is chosen to d

_{step}=d

_{pix}·N/(N+1). This results in measurement positions which are uniformly distributed in-between the reconstruction positions x

_{k}, except for the boundaries (cf. Fig. 3).

## 4. Simulation results

_{pix}=189 µm). Measurement noise of the distance sensors (5 nm), autocollimator (0.2 arcsec (1 µrad)), distance interferometer (250 nm), positioning errors (5 µm) and tilt errors (2 arcsec (10 µrad)) of the scanning stage have been accounted for; the figures in brackets indicate the standard deviations of the (normally distributed) random variables used to simulate the various error sources. Further on, the single distance sensors have been simulated with an effective sensitive area of d

_{width}=17.96 µm leading to an averaging of the topography over a corresponding width. For the reconstruction procedure, the reconstruction distance was set to d

_{s}=d

_{pix}and the degree of the interpolation polynomial to o=41. The reconstruction algorithm described in the previous Section allows for a reconstruction only up to an arbitrary straight line. On this account, the reconstructed topography has been rotated and shifted relative to the desired topography such that the root mean square error is minimal. As test functions sinusoidal topographies with varying wavelengths and amplitude of 100 nm have been used. Since we are interested in studying aliasing effects, topography reconstruction errors were assessed as follows: For topographies consisting of frequencies smaller than the Nyquist frequency (f

_{Nyq}=1/2d

_{s}), the usual root mean square error was used. For topographies consisting of higher frequencies the root mean square error to a flat topography was calculated, as perfect anti-aliasing would yield a flat reconstructed topography in these cases. Figure 4 shows the obtained results for sparse (d

_{step}=d

_{pix}) and for dense (d

_{step}=d

_{pix}·N/(N+1)) sampling.

## 5. Measurement results

12. B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE **5878**, 587806 (2005).
[CrossRef]

^{−4}down to λ=19 µm to obtain higher resolution. Figure 5 shows the wavelengths of the chirp specimen as function of the distance to the center.

8. µPhase Interferometer, FISBA Optik AG, CH-9016 St. Gallen and FISBA Optik GmbH, Berlin, http://www.fisba.ch/.

14. A. Wiegmann, C. Elster, M. Schulz, and M. Stavridis, “Absolute Topographievermessung gekruemmter optischer Oberflaechen mit hoher lateraler Aufloesung”, in *Proceedings of the 109th DgaO*, http://www.dgao-proceedings.de/download/109/109_p28.pdf.

^{th}pixel, thereby enlarging the pixel distance by factor of 17 which models the measurements by another distance sensor array having the larger pixel distance d

_{pix}=323 µm and N=8 pixels. Since the reconstruction distance is chosen equal to the pixel distance (d

_{s}=d

_{pix}), reconstruction of wavelengths down to 646 µm is possible. Hence aliasing can be expected in a significant part of the designed chirp specimen (cf. Fig. 5). By comparing the results obtained for this reduced sensor array with those obtained for the original measurements we can assess the amount of aliasing induced by the former for wavelengths which can be reconstructed by the original measurements but not by those of the reduced sensor array.

_{s}=21 µm) serving as reference for the second using dense sampling (d

_{s}=323 µm).

_{s}=646 µm, hence aliasing effects are expected to occur beyond these lines for sparse sampling.

_{s}(marked by the red vertical line). For the sparse sampling (d

_{s}=d

_{pix}, Fig. 7) the reconstructed profile is jumping up and down on the right hand side of the red line showing significant aliasing.

^{th}pixel of the data used for constructing the orange reference curve. Hence profile reconstruction with a high lateral resolution and subsequent low pass filtering is not possible to obtain the same results as in Fig. 8.

_{topo}/f

_{Nyq}=2M (M=1,2,3,…) lead to rms reconstruction errors larger than the amplitude of the specimen also for dense sampling. At the position x=214.4 mm the chirp has a local frequency of f

_{topo}/f

_{Nyq}=2 and at x=221.05 mm local frequency of f

_{topo}/f

_{Nyq}=4. But perhaps since the frequency of the chirp is continuously varying, aliasing effects for these singular frequencies appear to have only a minor effect.

## 6. Conclusion

## Acknowledgments

## References and links

1. | L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. |

2. | L. Lahousse, S. Leleu, J. David, O. Gibaru, and S. Ducourtieux, “Z calibration of the LNE ultra precision coordinate measuring machine,” in |

3. | J. Cohen-Sabban and D. Reolon, “Vibration insensitive 3D-profilometry: a new type of white light interferometric microscopy” Proc. SPIE 7064, (2008). [CrossRef] |

4. | D J Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E Sci. Instrum. |

5. | W. Gao and S. Kiyono, “High accuracy profile measurement of a machined surface by the combined method,” Measurement |

6. | C. Elster, I. Weingaertner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. |

7. | A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express |

8. | µPhase Interferometer, FISBA Optik AG, CH-9016 St. Gallen and FISBA Optik GmbH, Berlin, http://www.fisba.ch/. |

9. | MPLS 180, STIL - 595, rue Pierre Berthier - Domaine de Saint Hilaire - 13855 Aix en Provence Cedex 3 - FRANCE, http://www.stilsa.com/. |

10. | E. Marsh, J. Couey, and R. Vallance, “Nanometer-Level Comparison of Three Spindle Error Motion Separation Techniques,” J. Manuf. Sci. Eng. |

11. | V. Bakshi, |

12. | B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE |

13. | R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete”. Technisches Messen |

14. | A. Wiegmann, C. Elster, M. Schulz, and M. Stavridis, “Absolute Topographievermessung gekruemmter optischer Oberflaechen mit hoher lateraler Aufloesung”, in |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(100.3008) Image processing : Image recognition, algorithms and filters

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 8, 2009

Revised Manuscript: June 7, 2009

Manuscript Accepted: June 11, 2009

Published: June 18, 2009

**Citation**

Axel Wiegmann, Michael Schulz, and Clemens Elster, "Suppression of aliasing in multi-sensor scanning absolute profile measurement," Opt. Express **17**, 11098-11106 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-11098

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### References

- L. Deck and P. de Groot, "High-speed noncontact profiler based on scanning white-light interferometry," Appl. Opt. 33, 7334-7338 (1994). [CrossRef] [PubMed]
- L. Lahousse, S. Leleu, J. David, O. Gibaru, S. Ducourtieux, "Z calibration of the LNE ultra precision coordinate measuring machine," in Proceedings of the 9th international conference of the european society for precision engineering and nanotechnology, V 2, 348-353 (2007).
- J. Cohen-Sabban and D. Reolon, "Vibration insensitive 3D-profilometry: a new type of white light interferometric microscopy," Proc. SPIE 7064, 706405-706405-9 (2008). [CrossRef]
- D. J. Whitehouse, "Some theoretical aspects of error separation techniques in surface metrology," J. Phys. E. Sci. Instrum. 9, 531-536 (1976). [CrossRef]
- W. Gao and S. Kiyono, "High accuracy profile measurement of a machined surface by the combined method," Measurement 19, 55-64 (1996). [CrossRef]
- C. Elster, I. Weingaertner, and M. Schulz, "Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors," Prec. Eng. 30, 32-38 (2006). [CrossRef]
- A. Wiegmann, M. Schulz, and C. Elster, "Absolute profile measurement of large moderately flat optical surfaces with high dynamic range," Opt. Express 16, 11975-11986 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975. [CrossRef] [PubMed]
- µPhase Interferometer, FISBA Optik AG, CH-9016 St. Gallen and FISBA Optik GmbH, Berlin, http://www.fisba.ch/.
- MPLS 180, STIL - 595, rue Pierre Berthier - Domaine de Saint Hilaire - 13855 Aix en Provence Cedex 3 - FRANCE, http://www.stilsa.com/.
- E. Marsh, J. Couey, and R. Vallance, "Nanometer-Level Comparison of Three Spindle Error Motion Separation Techniques," J. Manuf. Sci. Eng. 128, 180-187 (2006) [CrossRef]
- V. Bakshi, EUV Lithography (John Wiley & Sons, 2009), Chap. 5.3
- B. Doerband and J. Hetzler, "Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF)," Proc. SPIE 5878, 587806 (2005). [CrossRef]
- R. Krueger-Sehm, P. Bakucz, L. Jung, H. Wilhelms, "Chirp-Kalibriernormale fuer Obeflaechenmessgeraete," Technisches Messen 74,572-576 (2007).
- A. Wiegmann, C. Elster, M. Schulz, M. Stavridis, "Absolute Topographievermessung gekruemmter optischer Oberflaechen mit hoher lateraler Aufloesung," in Proceedings of the 109th DgaO,http://www.dgao-proceedings.de/download/109/109_p28.pdf.

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